Page:Calculus Made Easy.pdf/51

it is, however, always worth while to try whether the expression can be put in a simpler form.

First we must try to bring it into the form $y=$ some expression involving $x$ only.

The expression may be written

$(a-b)y+(a+b)x=(x+y){\sqrt {a^{2}-b^{2}}}$.

Squaring, we get

$(a-b)^{2}y^{2}+(a+b)^{2}x^{2}+2(a+b)(a-b)xy$

⁠$=(x^{2}+y^{2}+2xy)(a^{2}-b^{2})$,

which simplifies to

$(a-b)^{2}y^{2}+(a+b)^{2}x^{2}=x^{2}(a^{2}-b^{2})+y^{2}(a^{2}-b^{2})$;

or

$\left[(a-b)^{2}-(a^{2}-b^{2})\right]y^{2}=\left[(a^{2}-b^{2})-(a+b)^{2}\right]x^{2}$,

that is

$2b(b-a)y^{2}=-2b(b+a)x^{2}$;

hence $y={\sqrt {\frac {a+b}{a-b}}}x$ and ${\frac {dy}{dx}}={\sqrt {\frac {a+b}{a-b}}}$.

(4) The volume of a cylinder of radius $r$ and height $h$ is given by the formula $V=\pi r^{2}h$. Find the rate of variation of volume with the radius when $r=5.5$ in. and $h=20$ in. If $r=h$, find the dimensions of the cylinder so that a change of $1$ in. in radius causes a change of $400$ cub. in. in the volume.

The rate of variation of $V$ with regard to $r$ is

${\frac {dV}{dr}}=2\pi rh$.

If $r=5.5$ in. and $h=20$ in. this becomes $690.8$. It means that a change of radius of $1$ inch will cause a change of volume of $690.8$ cub. inch. This can be easily verified, for the volumes with $r=5$ and $r=6$